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We develop a general criterion about coarsening for a class of nonlinearevolution equations describing one dimensional pattern-forming systems. Thiscriterion allows one to discriminate between the situation where a coarseningprocess takes place and the one where the wavelength is fixed in the course oftime. An intermediate scenario may occur, namely `interrupted coarsening'. Thepower of the criterion lies in the fact that the statement about the occurrenceof coarsening, or selection of a length scale, can be made by only inspectingthe behavior of the branch of steady state periodic solutions. The criterionstates that coarsening occurs if lambda'(A)>0 while a length scale selectionprevails if lambda'(A)<0, where $lambda$ is the wavelength of the pattern and Ais the amplitude of the profile. This criterion is established thanks to theanalysis of the phase diffusion equation of the pattern. We connect the phasediffusion coefficient D(lambda) (which carries a kinetic information) tolambda'(A), which refers to a pure steady state property. The relationshipbetween kinetics and the behavior of the branch of steady state solutions isestablished fully analytically for several classes of equations. Anotherimportant and new result which emerges here is that the exploitation of thephase diffusion coefficient enables us to determine in a rather straightforwardmanner the dynamical coarsening exponent. Our calculation, based on the ideathat |D(lambda)|=lambda^2/t, is exemplified on several nonlinear equations,showing that the exact exponent is captured. Some speculations about theextension of the present results to higher dimension are outlined.Comment: 16 pages. Only a few minor changes. Accepted for publication in  Physical Review

Nonlinear dynamics in one dimension: A criterion for coarsening and its temporal law